Problem: What is the integer value of $\left((\sqrt{2})^{\sqrt{2}}\right)^{\sqrt{8}}$?
Answer: Recall that $(a^b)^c=a^{bc}$. Thus, the given expression is equivalent to $(\sqrt{2})^{\sqrt{2} \cdot \sqrt{8}}=(\sqrt{2})^{\sqrt{16}} =(\sqrt{2})^4$. Because $\sqrt{2}=2^{\frac{1}{2}}$, we can rewrite this as $(2^{\frac{1}{2}})^4=2^{\frac{1}{2} \cdot 4}=2^2=\boxed{4}$.